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Realization of Lie algebras and classifying spaces of crossed modules

Yves Félix and Daniel Tanré

Algebraic & Geometric Topology 24 (2024) 141–157
Abstract

The category of complete differential graded Lie algebras provides nice algebraic models for the rational homotopy types of nonsimply connected spaces. In particular, there is a realization functor, , of any complete differential graded Lie algebra as a simplicial set. In a previous article, we considered the particular case of a complete graded Lie algebra, L0, concentrated in degree 0 and proved that L0 is isomorphic to the usual bar construction on the Maltsev group associated to L0.

Here we consider the case of a complete differential graded Lie algebra, L = L0 L1, concentrated in degrees 0 and 1. We establish that the category of such two-stage Lie algebras is equivalent to explicit subcategories of crossed modules and Lie algebra crossed modules, extending the equivalence between pronilpotent Lie algebras and Maltsev groups. In particular, there is a crossed module 𝒞(L) associated to L. We prove that 𝒞(L) is isomorphic to the Whitehead crossed module associated to the simplicial pair (L,L0). Our main result is the identification of L with the classifying space of 𝒞(L).

Keywords
rational homotopy, realization of Lie algebras, Lie models of simplicial sets, crossed modules
Mathematical Subject Classification
Primary: 17B55, 55P62
Secondary: 55U10
References
Publication
Received: 8 March 2021
Revised: 22 August 2022
Accepted: 3 September 2022
Published: 18 March 2024
Authors
Yves Félix
Département de Mathématiques
Université Catholique de Louvain
Louvain-la-Neuve
Belgium
Daniel Tanré
Département de Mathématiques
Université de Lille
Lille
France

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